In this paper we explore the conditions under which the von Kries model of colour constancy is exact, an investigation motivated by the fact that in practice the model has been shown to work well for a wide range of imaging devices [1] despite the fact that existing theory [2] predicts that it should perform poorly. We present a modified theory which reconciles this apparent contradiction and which is based on the observation that von Kries adaptation treats sensor responses independently of one another. Starting from this point we show how to recover, for a single sensor, set of surfaces, and reference illuminant, the set of von Kries illuminants: all lights for which von Kries adaptation is a perfect model of illumination change. To help us in this task we use a linear model of surface reflectance, but importantly, we use a local model: that is, a model derived by examining reflectance only in the region to which the sensor of interest is sensitive. Adopting such a model and treating sensors independently of one another we show that our new theory accurately predicts the good practical performance of the von Kries model.